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An Investigation of Machine Learning Methods Applied

to Structure Prediction in Condensed Matter

William J. Brouwer1,a James D. Kubicki,b Jorge O. Sofo,c C. Lee Gilesd

aResearch Computing and Cyberinfrastructure, bDepartment of Geosciences, cDepartment

of Physics, dInformation Science and Technology, The Pennsylvania State University

Abstract

Materials characterization remains a significant, time-consuming undertaking.

Generally speaking, spectroscopic techniques are used in conjunction with em-

pirical and ab-initio calculations in order to elucidate structure. These exper-

imental and computational methods typically require significant human input

and interpretation, particularly with regards to novel materials. Recently, the

application of data mining and machine learning to problems in material science

have shown great promise in reducing this overhead [1]. In the work presented

here, several aspects of machine learning are explored with regards to character-

izing a model material, titania, using solid state Nuclear Magnetic Resonance

(NMR). Specifically, a large dataset is generated, corresponding to NMR 47Ti

spectra, using ab-initio calculations for generated TiO2 structures. Principal

Components Analysis (PCA) reveals that input spectra may be compressed by

more than 90%, before being used for subsequent machine learning. Two key

methods are used to learn the complex mapping between structural details and

input NMR spectra, demonstrating excellent accuracy when presented with test

sample spectra. This work compares Support Vector Regression (SVR) and Ar-

tificial Neural Networks (ANNs), as one step towards the construction of an

expert system for solid state materials characterization.

1Corresponding author, email address : wjb19@psu.edu

Preprint submitted to Elsevier May 15, 2014

1. Introduction

Structure characterization is one of many interesting problems in material

science, the latter recently garnering significant attention in the form of the

Materials Genome Initiative [4, 5], which ultimately seeks to understand the

atomistic blueprint for key materials. Researchers across various scientific disci-

plines seek to develop structural models for condensed and molecular systems.

The modeling process revolves around the gradual refinement of assumptions,

through comparison of experimental and computational results. A critical ex-

perimental technique used in material science is solid-state NMR, a method that

provides great insight into chemical order over Angstrom length scales, an im-

portant spectroscopic tool used in key discoveries [2]. However, interpretation

of spectra for new and complex solid-state materials is difficult, often requiring

experiments on model compounds in order to derive empirical relationships, for

interpretation of the system under study [3], overall a time-consuming process.

Spectra must also generally be simulated and fit in order to extract parameters

that correspond to structural features.

This process of simulation and fitting is common to many experimental tech-

niques, including X-ray spectroscopy. Similarly, working forward from structural

models in order to produce measurable experimental quantities calculated from

first principles is computationally demanding. Thus, the process of structure

determination has significant impediments, slowing the time to discovery sig-

nificantly. The present work explores the introduction of machine learning to

materials characterization, specifically to quantify structural distortions in sim-

ple oxides, which has relevance to characterizing more complex oxides including

Relaxor Ferroelectrics [6].

The application of machine learning is common to several disciplines, for

instance, recent work has been devoted to creating a method to predict the out-

come of chemical reactions in organic chemistry, provided with input reactants

and conditions[7]. Expert systems have been developed for structure deter-

mination of organic molecules, from acquired crystallographic and NMR data

2

[8, 9]. Similarly, a variety of approaches have been devised for determining pro-

tein structure from 2D NMR experiments[10]. Also within structural biology,

machine learning has been applied to predict protein-ligand binding affinity[11].

These chemical or molecular examples stand in distinction to the charac-

terization of condensed or solid-state materials, where methods of calculating

fundamental quantities and spectroscopic observations present different chal-

lenges. For example, although protein structure is complicated and nuanced,

2D and 3D NMR spectra of proteins are generally composed of well-resolved

lines that have direct correlations with structural details such as bond lengths

and chemical species. On the other hand, solid-state spectra for condensed ma-

terials generally suffer from line broadening mechanisms to be discussed shortly

that degrade spectral resolution and make interpretation difficult. This degra-

dation in resolution is made worse by local disorder in bonding environments,

for example in glasses and solid solutions. Similar difficulties pervade other

spectroscopic techniques including X-Ray diffraction. With regards to first-

principle calculations of spectroscopic quantities, molecular systems can prove

formidable but manageable computationally. Gaussian-based orbitals have been

used for many decades in the solution of the Schrodinger equation for molecular

systems [12], and in conjunction with approximations for electronic exchange

and correlation effects, provide accurate values for a wide variety of measurable

quantities. On the other hand, calculations of electronic structure in extended

systems require the use of periodic boundary conditions and plane wave orbitals

for the electronic states, which is much less computationally tractable, for rea-

sons to be discussed shortly. Nonetheless, great strides have been made in the

development and use of ab-initio calculations in solid-state material science.

The use of Density Functional Theory (DFT) [13] in particular has increased

dramatically over the last decade, permitting scientists to evaluate potentially

useful materials computationally [14, 15], without the need for costly synthesis

and spectroscopy. As a computational tool, DFT also allows for the study and

structure determination of inaccessible materials eg., the inner core of terrestrial

planets [16].

3

1.1. Contribution

Machine learning is generally used in order to derive useful information and

relationships, particularly for large datasets. Machine learning exists in many

forms, although all methods may generally be regarded as being either unsuper-

vised or supervised in nature. Supervised methods are those that take a number

of data examples during a training phase, where the input and output spaces for

the data can be widely varying in size and nature. During the training phase, a

mapping between the two spaces is determined and represented in a model ger-

mane to the method used, such that when presented with new data, predicted

output is returned for a given input. Predictions might be binary by nature

i.e., classification or numerical i.e., regression, or a combination. Attention in

this work has been restricted to exploring the use of machine learning meth-

ods for regression, a computationally attractive and well established technique.

Features are extracted from normalized NMR spectra, generated from ab initio

calculation and simulation based on model structures; output values correspond

to unit cell parameters used in computations. Model structures are generated

by randomly permuting unit cell parameters, rejecting those candidates whose

interatomic distances violate steric considerations. In other words, should bond

lengths be less than the sum of excepted ionic radii, candidates are rejected.

This work examines both Multiple-Input, Multiple-Output Support Vector

Regression (MSVR) and Artificial Neural Networks (ANNs), comparing compu-

tational time, scaling and accuracy of methods, when used to discern the map-

ping between input spectra and unit cell parameters of materials that give rise

to spectra. When presented with the simulated spectra for a related polymorph

of the structures used during the training phase, both methods reproduce most

unit cell parameters fairly accurately. The overall approach should be amenable

to other form of materials data, which in conjunction with suitably constructed

and arranged machine learning elements would comprise an expert system for

solid state materials characterization, for the elucidation of complex materials.

4

2. Theory

2.1. Density Functional Theory

The following serves only as a brief review of DFT, which approximates the

many body Schrodinger equation for N electrons, in terms of an interaction

between a single electron (with wavefunction ψi and energy ǫi) and a charge

density :

[

−~2

2m∇2 + V (r)

]

ψi(r) = ǫiψi(r) (1)

where the charge density n(r) is given by :

n(r) =

N∑

i

|ψi(r)|2 (2)

The effective potential comprises three terms :

V (r) = Vext(r) + Vee(r) + Vxc(r ) (3)

where Vext(r) is the external potential, and Vee(r) and Vxc(r) are the electron-

electron repulsion and exchange-correlation contributions, both functionals of

the density. A variety of approximations have been devised over the years for

the latter, two common approaches are the Local Density Approximation (LDA)

and the Generalized Gradient Approximation (GGA). The solution to equation

1 is obtained via a self-consistent iterative process, whereby approximations to

ground-state wave functions are produced after convergence. In extended solids,

the most popular basis for the expansion of these states is plane-waves that

in conjunction with equation 1 gives rise to a generalized eigenvalue problem.

The excessive number of plane waves required in this expansion for non-valence

electronic states prompted the creation of pseudopotentials, where the rapidly

oscillating wavefunction near the core is replaced by pseudized, smoothed ap-

proximations with fewer nodes.

5

2.2. Nuclear Magnetic Resonance

Once ground-state wavefunctions have been deduced, one may calculate mea-

surable quantities for the material, for example, NMR parameters. The most

significant interaction in NMR is between the nuclear magnetic moment and an

applied static magnetic field (Zeeman effect), producing the Larmor precision at

frequency ω0, which is observable at radio frequencies [19]. In both liquid and

solid state NMR, shifts to these frequencies are produced by the interaction be-

tween induced electronic currents and nuclear magnetic moment, the chemical

shift interaction. The dipole interaction between magnetic moment of different

nuclei produces significant line broadening, reduced to a large degree in the

liquid state by the tumbling motion of molecules. Indeed, rapid interpretation

of liquid state NMR spectra has been a routine analytic technique in organic

chemistry for many decades. In liquid state NMR, lines in the frequency spec-

trum are generally well resolved and chemical shifts are directly correlated with

bond lengths and chemical species, particularly for organic materials. In terms

of magnitude, the most significant interaction besides the chemical shift for rele-

vant nuclei is the quadrupole interaction, a function of the electric field gradient

V at the nucleus, whose components contribute to the measurable quadrupole

coupling constant CQ and asymmetry parameter η:

CQ =eQVzzh

; η =Vyy − Vxx

Vzz(4)

Only nuclei with spin I > 1/2 including isotopes 47Ti (I = 5/2) and 49Ti (I

= 7/2) have a non-zero quadrupole moment Q, which couples with the electric

field gradient. This introduces anisotropic line broadening to the NMR spectra

of powdered solids composed of many crystallite orientations α, β distributed

over the unit sphere. Materials are frequently studied in this form, given the

difficulty of synthesizing single crystal examples. The associated line broadening

greatly complicates interpretation; lines for distinct chemical sites overlap and

simulation is necessary in order to extract parameters such as CQ and η in the

6

solid state. Magic Angle Spinning (MAS) is an experimental technique that

alleviates broadening, reducing or removing first order effects; to second order,

average Hamiltonian theory gives for the quadrupole frequencies [20]:

ωr,c = −r − c

ω0

[

CQ

2I(2I + 1)

]2 {

A(0)(I, r, c)

(

η2 + 3

10

)

+

A(4)(I, r, c)f(η, α, β)}

(5)

These frequencies are a function of the energetic transition r ↔ c, crystallite

orientation α, β, spin I and aforementioned quadrupole parameters. For the

purposes of this work, attention is restricted to this interaction. Figure 1 pro-

vides two examples of simulations using equation 5, for the central frequency

transition (r, c = 1/2,−1/2) measured in common experiments.

2.3. Support Vector Regression

The application of neural networks to regression is well documented, partic-

ularly in finance [21, 22], however the adaptation of the support vector approach

to regression is more recent. The linear regression problem is commonly stated

as solving for w,b in :

y = f(x) = 〈w,x〉+ b (6)

where xi, yi are input-output data pairs. The solution process minimizes the

norm, under the assumption that the optimal solution approximates all data

pairs with precision ǫ. It may be shown that a solution to the optimization

problem leads to a Lagrangian formulation, whose solution for the weights w is

a linear function of input data, the support vector expansion [24, 23]. In general,

the relationship between input (feature) and output (value) space is non-linear,

and kernels φ are applied in order to map features to a higher dimensional space,

in order to maintain a solution form comprising a linear combination of support

7

Figure 1: Simulated 47Ti MAS lineshapes for a) a material with quadrupole coupling con-

stant of 1MHz and assymetry parameter of 1.0 and b) a material with quadrupole coupling

constant of 5Mhz and assymetry parameter of 0, Larmor frequency of 50.75 MHz and 5.12

kHz bandwidth.

8

vectors and features φ(x). Support vector regression has been generalized fur-

ther to solve multiple-input multiple-output (MIMO) problems, abbreviated in

this work as MSVR [25]. In this technique, the following Lagrangian expression

is minimized:

LP (W,b) =1

2

M∑

j=1

||wj2||+ Cn∑

i=1

L(ui) (7)

where C is a constant analogous to the soft margin parameter, W,b are now

multi-dimensional regression parameters to be determined,

W = [w1, ...,wM ];b = [b1, ...,bM ]T (8)

and L(ui) is defined as :

L(ui) =

0, ui < ǫ

u2i − 2uiǫ+ ǫ2, ui ≥ ǫ(9)

Equation 9 is an expression of the penalty for predictions lying outside the

desired precision ǫ, a function of feature-value combinations {xi,yi} and trans-

formation kernels φ as follows:

ui = ||ei|| =√

eTi ei

eTi = yTi − φT (xi)W − bT (10)

The training set consists of i = 0, ..., n examples, and x,y have dimensions

N,M respectively. The scaler ui is distinct for each input example, the norm

of vector ei, in turn a function of output yi, features φ(xi) and regression

parameters. The minimization procedure is a non-linear problem solvable in

9

an iterative fashion, and therefore an expression for the gradient is required.

This is developed from the aforementioned equation using a Taylor expansion,

ultimately leading to the solution of a linear system, for each step of the iterative

procedure. After regression parameters have been deduced, one may develop

predictions for new input x’:

y′ = φT (x′)ΦTβ, (11)

using an expansion for weights as a function of training input x and parameters

βj optimized by the iterative procedure :

wj =∑

i

φ(xi)βj = ΦTβj , (12)

the multiple-output analog of the support vector expansion for single-output

data.

3. Experiments

3.1. Methods

As alluded to previously, the main goal of this work is to determine the ef-

ficacy of machine learning methods in deriving structural details directly from

input solid state NMR spectra. Therefore, both DFT computations from model

structures, and corresponding NMR simulations are required, in order to cre-

ate input features (from spectra) and output values (unit cell parameters used

in DFT calculations). Titania (titanium oxide) was chosen as the material for

this work, owing to it’s industrial and environmental relevance and wealth of

available published information [26, 27], table 1. Programs from within the

Quantum Espresso [28] suite were used for the pseudopotential (ld1.x), DFT

(pw.x) and NMR parameter calculations (gipaw.x). Simulations of the mea-

surable 47Ti NMR spectrum were performed using custom software [29], and

10

the method detailed in the previous section was coded in C++ for the machine

learning (regression) steps, using support vectors. Before proceeding with the

batch process to be outlined shortly, pseudopotentials for Ti and O were gen-

erated using the Perdew-Burke-Ernzerhof [30] exchange-correlation functional,

capable of calculating magnetic response (NMR) parameters using the Gauge

Including Projector Augmented Wave (GIPAW) [31] Method. All DFT calcu-

lations were performed using a mesh of 4 × 4 × 4 k-points in order to sample

the Brillouin zone, ultimately providing values of quadrupole coupling constant

and asymmetry parameter for rutile and anatase in good agreement with ex-

perimental values [32]. The following batch process was performed, beginning

with the structure for Anatase :

1. Generate TiO2 structure with fixed Ti coordinates and angles throughout,

independently perturb O fractional coordinates x ≡ y, z and unit cell

parameters a ≡ b, c, with random displacements.

2. If new atomic positions violate steric considerations (ie., distance between

Ti4+ and O2− is less than sum of ionic radii = 2 A), reject the move, else:

3. Calculate ground state electronic structure for system via DFT

4. Calculate 47Ti quadrupole coupling constants and asymmetry parameters

5. Simulate 47Ti MAS NMR spectrum composed of 512 amplitude points,

using a Larmor frequency of 50.75 MHz and 5.12 kHz bandwidth.

Therefore, the output value space is of dimension 4 (cell parameters Ox ≡

Oy, Oz, a ≡ b, c) and input feature space is of dimension 512. Principle Com-

ponents Analysis (PCA) was used in this work to compress the input data

space, in order to expedite the training process. Referring to the input data

as rectangular matrix X , with n rows corresponding to different experiments

and N = 512 columns, PCA proceeds by first finding the eigenvalues of X ′X .

This is generally accomplished by computing the more tractable singular value

decomposition :

X = UΣW ′ (13)

11

where the non-zero elements of Σ, the singular values, correspond to the square

roots of the eigenvalues λ of X ′X and W ′ corresponds to the eigenvectors of

X ′X . By retaining the N ′ largest eigenvalues and corresponding eigenvectors

in columns of matrix P , X may be transformed to a smaller space n×N ′ using

the transformation Y = X ∗ P .

Table 1: Key polymorphs of titania; α, β, γ = 90, Ti=(0,0,0)

Formula Name Group a, b c Ox, Oy Oz

TiO2 Rutile P42/mnm 4.5922 2.9574 0.30496 0

TiO2 Anatase I41/amd 3.7842 9.5146 0 0.20806

3.2. Results

By generating data after the fashion described, a region within a four di-

mensional output parameter space is effectively explored. This example benefits

from the relatively high symmetry of titania and associated reduction in com-

putation time. Over 1000 experiments as described prior were performed using

eight Intel Sandy Bridge processors, in under twelve hours. A dataset was ex-

tracted from experiments, with inputs and outputs as described, by thresholding

and selecting the first 500 elements with quadrupole coupling constant ≤ -0.95

MHz. The initial dataset contained both positive and negative values for CQ,

and measurable second order quadrupole frequencies are insensitive to the sign

of CQ. Figure 2 displays the NMR parameters for the selected dataset.

After thresholding, the effects of compression on training time and accuracy

were investigated. Figure 3 displays a comparison between training time for the

nnet package within R, used to implement an ANN, and the aforementioned

MSVR method, as implemented in C++. Referring to this figure, as expected,

MSVR scales almost linearly with input dimension, while the ANN scales poorly.

Training time for the latter increases exponentially with input data dimension,

and at input dimension of 128, exceeded the limitations of the package (1000

weights). In order to asses accuracy during experiments, the Root Mean Squared

12

Figure 2: NMR parameters for dataset; structures with larger magnitude CQ and smaller η

are similar to anatase, while structures with smaller magnitude CQ and larger η are similar

to rutile.

Error (RMSE) was used :

∆ =

n∑

i

√

(y′i − yi)2

n(14)

where y′i are values predicted by the particular machine learning method em-

ployed, yi are the real values, and the sum is carried out over the number of

test examples n in the datafold.

It was observed that both methods maintain a high degree of accuracy with

compression (figure 3b), an encouraging result with regards to establishing a

practical database. The difference in total RMSE for both methods was ap-

proximately 5%, between using an input dimension of 2 and 64 features. This is

due in no small part to the simplicity of the system and thus spectra; in a more

practical situation spectra are much more feature rich, for example in the pres-

ence of multiple, overlapping spectral lines (inequivalent chemical sites). For the

13

Figure 3: Results of scaling studies : a) time required to train the ANN (solid line) and MSVR

(circles) in seconds, versus the log of the input dimension b) the total RMSE of the ANN as

a function of the log of the input dimension.

remainder of experiments, the input data dimension was compressed to 4 input

features. Both methods were compared using ten-fold cross validation, in order

to assess overall accuracy and costs of parameter optimization. The RMSE data

for both methods is recorded in tables 2 and 3; clearly both methods have com-

parable error. However, the ANN for small input data dimensions is optimized

fairly quickly (56 weights, 4-4-4 network) in distinction to MSVR, which at this

stage requires tuning of C and kernel function parameters for a desired precision,

in addition to the optimization procedure for regression variables W,b. With

regards to MSVR, the only kernel functions that produced reasonable output

were radial basis functions (with optimal γ in the range 0.01 to 0.1); linear and

polynomial kernels produced far more inaccurate predictions. Overall, these

machine learning methods do show that solid-state NMR provides a sensitive

measure for certain unit cell parameter displacements. Figure 4 shows the dis-

tributions of relative error |1 − y′i/yi| from all data folds (10×50 = 500 test

14

Figure 4: Distribution of relative errors using MSVR prediction : a) Ox, Oy fractional co-

ordinate b) Oz fractional coordinate c) unit cell parameter a, b and d) unit cell parameter

c.

samples). Fractional coordinate Oz, parameters a ≡ b and c are predicted to

within less than 10% relative error, 51%, 93% and 44% of the time respectively,

while predictions for Ox ≡ Oy are wildly inaccurate in many instances. The

fractional coordinates for O in x, y dimensions were observed to have less than

10% relative error in less than 13% of instances. This points to limitations in

using a single NMR interaction and indeed single spectroscopic data source in

making accurate predictions for all unit cell parameters. A larger expert system

would of course incorporate more NMR interactions (including chemical shift)

and other data sources, for instance X-ray spectra.

4. Conclusions

This work details several aspects of building a larger expert system for solid

state physics, demonstrating the use of MSVR and ANNs in learning the map-

15

Table 2: MSVR results; RMSE for ten data folds (450 train, 50 test elements, ǫ=0.2)

∆(Ox ≡ Oy) ∆(Oz) ∆(a ≡ b) ∆(c) C

0.063600 0.045500 0.243000 1.055900 5.600000

0.073900 0.045000 0.239500 1.461700 6.000000

0.077300 0.054400 0.240800 1.396600 4.600000

0.062000 0.040200 0.252300 1.255500 5.000000

0.066600 0.043100 0.224400 1.147100 6.000000

0.055800 0.045000 0.236700 1.389000 6.000000

0.079200 0.045600 0.268100 1.479500 5.000000

0.060100 0.057300 0.241400 1.231500 3.200000

0.072600 0.045500 0.303200 1.245300 2.600000

0.082800 0.049300 0.227800 1.261700 3.000000

ping between spectra and structure for model systems. These systems are gen-

erated using fixed composition but variable atomic positions. The virtue of

this approach is that more complicated materials, for example oxide surfaces,

are composed of simpler systems albeit with unknown atomic coordinates. By

repeating the process outlined here for more models, the aim is to create a

database and expert system for the elucidation of materials including oxide

surfaces, composed of simpler systems.

In a complete expert system for solid state materials, machine learning ele-

ments may be trained on various types of spectroscopic data for known struc-

tures, so that spectra for new materials may be presented to the system and

underlying structure deduced rapidly. The contribution presented here is predi-

cated on knowledge of the underlying chemical objects comprising the material.

In the absence of this knowledge, a process of classification or unsupervised

learning must take place first. Also, more complex systems including solid so-

lutions generally require large super cells in order to accurately perform DFT

calculations of measurable parameters, exponentially increasing the dimension-

ality of the feature space that must be explored, in order to produce reliable

16

Table 3: ANN results; RMSE for ten data folds (450 train, 50 test elements)

∆(Ox ≡ Oy) ∆(Oz) ∆(a ≡ b) ∆(c)

0.061488 0.039988 0.239628 1.187439

0.069671 0.044444 0.219666 1.403309

0.070845 0.053725 0.236521 1.500705

0.061350 0.038591 0.237223 1.303657

0.067755 0.042339 0.219838 1.172234

0.057470 0.043959 0.221471 1.314616

0.074059 0.046857 0.228791 1.488244

0.066621 0.049576 0.235042 1.190761

0.072513 0.039145 0.228004 1.339722

0.079725 0.045298 0.225725 1.436712

models. However, while not addressed in this work, many candidate structures

can potentially be ruled out on energetic and other grounds. Before a DFT cal-

culation for a condensed system can proceed, pseudopotentials for constituent

atoms must exist, constructed using the same exchange-correlation functional

to be applied to the extended system under study. As mentioned, a pseudopo-

tential drastically reduces the number of plane waves required in a calculation,

by using an approximation to the core region of the potential experienced by

electrons. Considerations as to the suitable partition of valence and core or-

bitals for a given environment strongly dictates the success or lack thereof in

using pseudopotentials. As a rule of thumb, explicitly including more valence

electrons provides greater transferability to different bonding environments, at

the expense of computation time.

In order to build an expert system, a survey and compilation of appropri-

ate pseudopotentials would need to be performed, with particular emphasis on

the ability to reproduce measurable quantities such as those used in this work.

Finally, parameters such as quadrupole coupling constants in NMR are propor-

tional to tensor traces ie., are insensitive to the sign on atomic displacements.

17

Nonetheless, augmented twith information from other iterations in NMR in-

cluding the chemical shift (particularly sensitive to chemical identity), or other

spectroscopic data, these limitations are readily overcome.

5. Acknowledgement

This work used resources on the TACC Stampede cluster, available via the

Extreme Science and Engineering Discovery Environment (XSEDE) initiative,

which is supported by National Science Foundation grant number ACI-1053575.

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